SOLITAIRE CLOBBER PLAYED ON HAMMING GRAPHS Paul Dorbec 1 ERTE Maths a Modeler, Institut Fourier, 100 rue des Maths, 38402 GRENOBLE, France paul.dorbec@ujf-grenoble.fr Eric Duchˆ ene 2 ERTE Maths a Modeler, Institut Fourier eric.duchene@ujf-grenoble.fr Sylvain Gravier 3 ERTE Maths a Modeler, Institut Fourier sylvain.gravier@ujf-grenoble.fr Received: 3/2/07, Revised: 4/8/08, Accepted: 4/11/08, Published: 4/17/08 Abstract The one-player game Solitaire Clobber was introduced by Demaine et al. Beaudou et al. considered a variation called SC2. Black and white stones are located on the vertices of a given graph. A move consists in picking a stone to replace an adjacent stone of the opposite color. The objective is to minimize the number of remaining stones. The game is interesting if there is at least one stone of each color. In this paper, we investigate the case of Hamming graphs. We prove that game configurationson such graphs can always be reduced to a single stone, except for hypercubes. Nevertheless, hypercubes can be reduced to two stones. 1. Introduction and Definitions We consider the one-player game SC2 that was introduced in [3]. This game is a variation of the game Solitaire Clobber defined by Demaine et al. in [2]. Note that both solitaire games come from the two-player game Clobber, that was created and studied in [1]. One can have a look to [4] for more information about Clobber. The game SC2 is a solitaire game whose rules are described in the following. Initially, black and white stones are placed on the vertices of a given graph G (one per vertex), forming what we call a game configuration. A move consists in picking a stone and ”clobbering” (i.e. 1 Universit Joseph Fourier 2 Postdoc in Universit de Lige 3 CNRS